We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$-dimensional families of manifolds of $K3^{[n]}$-type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.

We show that the possible jump of the order in an 1-parameter deformation family of (possibly nonisolated) hypersurface singularities, with constant Lê numbers, is controlled by the powers of the deformation parameter. In particular, this applies to families of aligned singularities with constant topological type—a class for which the Lê numbers are “almost” constant. In the special case of families with isolated singularities—a case for which the constancy of the Lê numbers is equivalent to the constancy of the Milnor number—the result was proved by Greuel, Plénat, and Trotman.

As an application, we prove equimultiplicity for new families of nonisolated hypersurface singularities with constant topological type, answering partially the Zariski multiplicity conjecture.

Although tangent bundles of manifolds are not always homotopically invariant, but in some categories of the manifolds they can be homotopically invariant.

In this note, I show that tangent bundles of π-manifolds and almost parallelizable manifolds depend only on their homotopy types.

We construct a discriminant-preserving map from the set of orbits in the space of quadruples of quinary alternating forms over the integers to the set of isomorphism classes of quintic rings. This map may be regarded as an analogue of the famous map from the set of equivalence classes of integral binary cubic forms to the set of isomorphism classes of cubic rings and may be expected to have similar applications. We show that the ring of integers of every quintic number field lies in the image of the map. These results have been used to establish an upper bound on the number of quintic number fields with bounded discriminant.

The Galois representation associated to a $p$-divisible group over a normal complete noetherian local ring with perfect residue field is described in terms of its Dieudonné display. As a consequence, the Kisin module associated to a commutative finite flat $p$-group scheme via Dieudonné displays is related to its Galois representation in the expected way.

In this paper, we give a characterization of the simultaneous unitarizability of any finite set of SL(2, ℂ)-valued functions on and determine all possible ways of the unitarization. Such matrix functions can be regarded as images of the generators for the fundamental group of a surface in an -family, and the results of this paper have applications in the construction of constant mean curvature surfaces in space.

Let K be a finite Galois extension of the rational number field Q and G a Gal(K/Q)-stable finite subgroup of GLn(OK). We have shown that G is of A-type in several cases under some restrictions on K. In this paper, we show that it is true for n = 2 without any restrictions on K.

Let S denote the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, G the Galois group of the quotient field extension, and f an element of Z2(G,U(S)) where U(S) denotes the multiplicative group of units of S. A crossed product Δ(f, S, G) whose radical is generated as a left ideal by the prime element II of S is an hereditary order according to the Corollary to Thm. 2. 2 of [2], and we call such a crossed product a II-principal hereditary order. In previous papers the author has studied II-principal hereditary orders Δ(f, S, G) for tamely and wildly ramified extensions S of R (see [10] and [11]). The purpose of this paper is to study II-principal hereditary orders Δ(f, S, G) with no restriction on the extension S of R.

In the classification problems of manifolds, the connected sums of sphere bundles over spheres appear frequently. In fact, the manifolds with certain tangential and homotopy properties come to such connected sums (cf. Tamura [15], [16], Ishimoto [6], [8], [9]). Motivated by those, in this paper and the subsequent paper, we classify connected sums of sphere bundles over spheres up to homotopy equivalence by extending the results of I. M. James and J. H. C. Whitehead [10], [11], which correspond to the case that the connected sums of the above happen to be single sums. We also use Wall [17] in the case when the fibres and the base spaces of bundles are same dimensional.

Un idéal entier I d’un corps quadratique réel k est dit primitif lorsque n ∈ N* et (n) divise I impliquent n = 1. Un idéal entier est primitif si et seulement si il vérifie les trois conditions suivantes:

(i) il n’est divisible par aucun idéal premier inerte,

(ii) il n’est divisible par le carré d’aucun idéal premier ramifié,

(iii) si il est divisible par un idéal premier totalement décomposé, alors il n’est pas divisible par son idéal premier conjugué.

For a finite-dimensional Lie algebra $\mathfrak {L}$ over $\mathbb {C}$ with a fixed Levi decomposition $\mathfrak {L} = \mathfrak {g} \ltimes \mathfrak {r}$ , where $\mathfrak {g}$ is semisimple, we investigate $\mathfrak {L}$ -modules which decompose, as $\mathfrak {g}$ -modules, into a direct sum of simple finite-dimensional $\mathfrak {g}$ -modules with finite multiplicities. We call such modules $\mathfrak {g}$ -Harish-Chandra modules. We give a complete classification of simple $\mathfrak {g}$ -Harish-Chandra modules for the Takiff Lie algebra associated to $\mathfrak {g} = \mathfrak {sl}_2$ , and for the Schrödinger Lie algebra, and obtain some partial results in other cases. An adapted version of Enright’s and Arkhipov’s completion functors plays a crucial role in our arguments. Moreover, we calculate the first extension groups of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules and their annihilators in the universal enveloping algebra, for the Takiff $\mathfrak {sl}_2$ and the Schrödinger Lie algebra. In the general case, we give a sufficient condition for the existence of infinite-dimensional simple $\mathfrak {g}$ -Harish-Chandra modules.

Let Γ be the unit circle and D be the open unit disk in the complex plane, and denote the Riemann sphere by Ω. By an arc at a point ζ∈Γ we mean a continuous curve such that |z(t)| < 1 for 0 ≦ t < 1 and . A terminal subarc of an arc Λ at ζ is a subarc of the form z = z (t) (t0 ≦ t < 1), where 0 ≦ t0<1. Suppose that f(z) is a meromorphic function in D. Then A(f) denotes the set of asymptotic values of f; and if ζ∈Γ, then C(f, ζ) means the cluster set of f at ζ and is the outer angular cluster set of f at ζ (see [13]).

Let C and C′ be two simple closed curves in the complex z-plane which have no point in common and surround the origin. Denote by D the annulus bounded by C and C′. Consider a family {γ} of rectifiable curves γ in D and the family P of all non-negative lower semi-continuous functions ρ = ρ(z) in D. Put

Let k be an algebraic number field of degree n = r1 + 2r2 with r1 real conjugates k(l) (1 ≦ l ≦ r1) and r2 pairs of complex conjugates k(m), k(m+r2)) (r1 + 1 ≦ m ≦ r1 + r2). Let o be the integral domain consisting of all integers in k.

It is interesting to know what kinds of primes are the factors of the class number of an algebraic number field, and especially to find ones being prime to the degree. About this matter it is desirable to construct the unramified Abelian extensions plainly. In this paper we shall show some of them for the prime cyclotomic number field and its quadratic extensions using the units of subfields.

In this article, we prove a finiteness result on the number of log minimal models for 3-folds in $\operatorname{char}p>5$. We then use this result to prove a version of Batyrev’s conjecture on the structure of nef cone of curves on 3-folds in characteristic $p>5$. We also give a proof of the same conjecture in full generality in characteristic 0. We further verify that the duality of movable curves and pseudo-effective divisors hold in arbitrary characteristic. We then give a criterion for the pseudo-effectiveness of the canonical divisor $K_{X}$ of a smooth projective variety in arbitrary characteristic in terms of the existence of a family of rational curves on $X$.

It is a well known fact on Lorenz groups that a quadratic form f is definite if and only if the corresponding orthogonal group On(R∞, f) where R∞ is the real number field, is compact. In this note, we shall show that the analogue of this holds for the case of the p-adic orthogonal group On(Rp, f), where Rp is the rational p-adic number field, as a special result of the more general statement on the completely valued fields.

In [1], Maass proved that the Dirichlet series associated with Siegel modular form satisfies a function equation. In this paper, we try to generalize the above fact to the indefinite case.

In a previous paper [4], we studied a class of algebras over a commutative ring R which we called semisimple algebras. Here we shall study simple algebras.

We study the asymptotic behavior in Sobolev norm of the local time of the d-dimensional fractional Brownian motion with N-parameters when the space variable tends to zero, both for the fixed time case and when simultaneously time tends to infinity and space variable to zero.